A364329 G.f. satisfies A(x) = (1 + x^3) * (1 + x*A(x)^2).
1, 1, 2, 6, 17, 52, 167, 558, 1912, 6683, 23736, 85426, 310861, 1141837, 4227938, 15764474, 59140089, 223062670, 845388258, 3217750229, 12295043520, 47144444476, 181349473833, 699629022954, 2706327445312, 10494497061015, 40787775234746, 158859378070721
Offset: 0
Programs
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Maple
A364329 := proc(n) add( binomial(2*n-6*k+1,k) * binomial(2*n-6*k+1,n-3*k)/(2*n-6*k+1),k=0..n/3) ; end proc: seq(A364329(n),n=0..70); # R. J. Mathar, Jul 25 2023
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Mathematica
nmax = 27; A[_] = 1; Do[A[x_] = (1 + x^3)*(1 + x*A[x]^2) + O[x]^(nmax+1) // Normal, {nmax+1}]; CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *) -
PARI
a(n) = sum(k=0, n\3, binomial(2*n-6*k+1, k)*binomial(2*n-6*k+1, n-3*k)/(2*n-6*k+1));
Formula
G.f.: A(x) = 2*(1 + x^3) / (1 + sqrt(1-4*x*(1 + x^3)^2)).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-6*k+1,k) * binomial(2*n-6*k+1,n-3*k) / (2*n-6*k+1).
D-finite with recurrence (n+1)*a(n) +2*(-2*n+1)*a(n-1) +(n+1)*a(n-3) +6*(-2*n+7)*a(n-4) +6*(-2*n+13)*a(n-7) +2*(-2*n+19)*a(n-10)=0. - R. J. Mathar, Jul 25 2023